Modeling the Thermal Conductivity of Concrete Based on Its Measured Density and Porosity

.loA. Tinker J.G. Cabrera

ABSTRACT material. In order to satisfactorily model the thennal conductivity of the composites from the properties of the The relationship between density and conductive heat individual components, it is necessary to choose a model transfer through mulliphase, porous materials is reasonably whose theoretical assumptions represent the distribution and well understood and can be evaluated using simple em­ shape of the component phases within the mixture as closely pirical relations. However, the complex effects ofporosity, as possible. However, in view of the complexity of the particularly pore size and volume, on heat transfer are structure of real materials, there are comparatively few mechanisms far less understood. To date, very few formulae cases where such theoretical assumptions can be applied are available that assess the effect that these variables have rigorously. The limiting factor in most models is the on heat conduction properties of a multiphase material. agreement between the theoretical assumptions and the This paper presents the results ofan experimental study structure of the actual material. designed to evaluate the influence ofpore characteristics on A small number of models have been reported that the thermal conductivity of concrete, a composite widely specifically calculate the conductive heat transfer of con­ used as a construction material. cretes (Pratt 1962; Valore 1980; Campbell-Allen and Twenty-one different concretes were made with densities Thome 1963). These models express the conductive heat that variedfrom 97to 146lbljii (1,550 to 2,350kg/m3) and transfer in terms of the conductivities and the fractional porosities from 10% to 39%. The thermal conductivity of amounts of the hydrated cement, the aggregate, and the the concretes was measured using a plain hot-plate techni­ pore phase. Only the conductive heat transfer of the que. Total porosity was determined using water and a entrained air is known with any certainty; the thennal vacuum saturation technique, and mercury intrusion conductivity of the solid component phase of the cement porosimetry was used to obtain the pore size distribution. paste and the aggregate are unknowns. Separate measure­ The experimentally derived heat transfer and porosity ments are particularly difficult to obtain and therefore the data were used to develop a mathematical model that techniques have a severely limited application. In view of reiates thermal conductivity to density, porosity, and median this, simplified expressions that predict the conductive heat pore diameter. The model predicts values of thermal transfer of composite materials using alternative parameters, conductivity that agree closely with experimental data. such as density and porosity, are of particular interest.

INTRODUCTION EXPERIMENTAL PROCEDURE AND RESULTS Knowledge of the thennal conductivity of concrete is SpecUication of the Materials important in many areas of construction, particularly to ensure the energy-efficient design of the exterior envelopes Quartz, limestone, and pellite were selected as coarse of buildings. The thermal conducting properties of concrete aggregates to investigate the effect of density and pore that are required are nonnall y measured under laboratory characteristics on the conductive heat transfer through conditions using, for example, a guarded or plain hot-plate concretes. Quartzitic river sand was used as a fme ag­ technique. Such procedures are time consuming and gregate. Quartz and limestone aggregates are natural expensive, and they require specially trained personnel and materials, while pellite is an artificial aggregate made by carefully prepared samples. pelletizing blast-furnace slag. Quartzitic coarse aggregate Numerous formulae have been developed over the years was used as a reference material. Limestone aggregate was that empirically predict the thermal conductivity of com­ chosen because it has nearly the same specific gravity and posite materials (Maxwell 1892; Eucken 1932; Brailsford porosity as quartz but a different mineral compositi.?n. and Major 1964; Reynolds and Hough 1957; Tinker 1987; pellite aggregate was selected because it has totally Simpson and Stuckes 1986). In most cases, the thermal characteristics, having a lower specific gravity and a por~II~, ; conductivity of a porous composite material depends on the and glassy matrix. conductivities of its component phases. the volume con­ Prior to any experimentation, all the centration of each. and the dispersion of the phases in the were air dried, sieved into single . John A. Tinker is a lecturer and J.G. Cabrera is a reader in the Department of Civil Engineering, Leeds

91 bined to obtain a particle size distribution within the range TABLE 1 recommended in British Standard 882, "Specification for Sieve Analysis for Coarse Aggregates Aggregates from Natural Sources for Concrete" (BSI 1986). The resultant blended grade that was used is given in Table BS 882 range for graded Blended 1. Sieve coarse aggre!!ates grade used. The fine aggregate, which consisted of a quartzitic river Size Percentage by mass Percentage passing BS sieves pa'\sing sand, underwent a similar preparation procedure. The British Standard grading range corresponding to zone M for 20mm 90 ~ 100 100.0 fine aggregates (BSI 882 1986) and the resultant size 14mm 54 ~ 75 74.8 IOmm 30 ~ 60 45.6 distribution are shown in Table 2. Sufficient fine pellite 5mm o ~ 10 5.0 aggregate was also prepared to the same particle size distribution so that the pore characteristics of a pellite TABLE 2 concrete could be studied. Sieve Analysis for Fine Aggregates

SpeeUication of the Mill: Designs BS 882 grading zone M for Blended Sieve fine a!!!!Telwtes grade used. Size Percentage by mass Percentage Mixes were designed so that the properties of the oassin!! BS sieves pa'\sing resultant concretes would represent a wide range of poro'si­ ties and densities. Variations in properties were achieved by 1.18mm 45 ~ 100 87.6 600~ 25 ~ 100 77.8 26.2 1. changing the cement-aggregate ratio, 300~ 5 ~ 48 2. changing the cement-sand ratio, 150lltn o ~ 10 5.0 3. changing the water/cement ratio, and 4 adding an air-entraining agent to selected mixes. TABLE 3 The composition of all the mixes is given in Table 3. Composition of the Concrete Mixes Mix Mix Type of Air- Total 'fest Methods No proportions Coarse Agg Entrained Water: (by ma~s) (0.26 litre Cement CementSand: per 100 kg ratio Measurement of Conductive Heat Transfer Twelve­ Agg cement) in.- (300-mm-) square, 2-in.- (50-mm-) thick concrete

specimens were cast and then stored in an environment 1 1 ,2.33,3.5 Quartzitic NO 0.53 maintained at 100% relative humidity and 68°F (20°C) for 2 Quartzitic YES 0.43 three days before being allowed to reach their equilibrium 3 Limestone NO 0.60 4 Lime.~tone YES 0.50 air-dry moisture content under ambient conditions of 65 % 5 Pellite NO 0.90 humidity and 68°F (20°C). Thermal conductivity measure­ 6 Pcllite YES O.RO ments were carried out according to BSI 874 (BSI 1988) 7 1 : 2.3"3:5.6 Quartzitic NO O.RO 8 Limestone NO 0.80 using an improved plain hot-plate apparatus designed and 9 Pellite NO 0.80 constructed at a British university. The apparatus consists 10 Quartzitic NO 0.50 of a central heater plate and two cold plates that sandwich 11 Limestone NO 0.66 12 Pellite NO I.OS the test samples. Unlike the guarded hot-plate, it has no 13 1 :3.73:5.6 Quartzitic NO O.9() guard ring surrounding the measurement area; however, the 14 Quartzitic YES O.RO edges of the spe(1imen are insulated. A correction is applied 15 Limestone NO 0.95 16 Limestone YES 0.90 to account for any heat loss that may take place from the 17 PeHilc NO 1.29 edges of the heater plate and specimens. The measurement 18 Pellite YES 1.29 uncertainty with this technique is ±5%, and the apparatus 19 Quartzitic NO 0.80 20 Limestone NO 0.80 was regularly calibrated against equipment in a British 21 'Pellite NO O.RO Calibration Service-accre.dited laboratory at a British university. One of the main sources of error when using the After completion of the measurement, the specimens unguarded hot-plate technique for measuring thermal were dried to a constant weight in a drying oven whose conductivity is the contact made b~tween the thermocouple's temperature was maintained at 221°F (105°C). Using the measurement junction and the concrete specimens. Intimate dry weight of the specimens, their dry density was deter­ contact was achieved by rolling the thermocouples, near mined. The conductive heat transfer values were adjusted to their hot junction, to a flatness of 0.0014 in. (0.035 mm) 3 % moisture content by volume using a derived moisture and applying a thin layer of heat sink compound between factor equation (Tinker et al. 1989). The results are pre­ the measurement junction and the specimen's surface. sented later in the paper.

92 TABLE 4 Measurement of Total Porosity The porosity of a Porosity Values Obtained by VacUlun Saturation material is the fraction of its bulk volume occupied by voids, and, in a material ,such as concrete or mortar, it can be determined by measuring any two of three quantities: Mix Mean Dry Tott'll Mean Total Porosity bulk volume, pore volume, or solid volume. In this inves­ No Density Porosity tigation, the porosity of the concrete was obtained from Ib/ft3 (kQ/m3) (%) (%) bulk and pore volume quantities (Ganjian 1990) that were I 143 (2282) 13.08 12.9 determined by vacuum saturation. 12.78 Essentially, vacuum saturation involves evacuating a 2 131 (2091 ) 13.89 14.0 pre-dried sample and then letting water fill the pores while 14.10 the sample is still under vacuum. During the test, air in the 3 141 (2248) 14.61 15.0 pores of the material is removed by negative pressure and 15.33 4 127 (2033) 20.91 21.0 is replaced by water. After a specific period, usually 21.16 between 24 and 36 hours, the pressure is increased back to 5 116 (1858) 21.44 21.0 atmospheric, during which time the sample reaches constant 20.63 weight. For this investigation, samples 3 in. (75 mm) in 6 97 (1557) 39.06 38.5 37.90 diameter were cored from 4-in. (lOO-mm) concrete cubes, 13.9 2 7 141 (2250) 14.09 and a vacuum pressure of about 13 Ib/in. (0.9 bar) was 13.69 used. Porosity was calculated using the following equation, 8 141 (2261) 13.86 14.1 and the results that were obtained are given in Table 4: 14.31 9 98 (1570) 23.42 23.4 W - Wd 23.33 P = s X 100 (1) Ws - Ww ' 10 146 (2341 ) 9.98 10.2 10.46 where 11 143 (2291 ) 11.84 12.1 12.32 (1647) 25.10 25.1 p = total porosity open to water, %; 12 103 25.05 = Ws saturated sample weight in air, Ib; 13 140 (2236) 15.38 15.1 Ww = saturated sample weight in water, lb; 14.91 Wd = oven-dried sample weight, lb. 14 126 (2020) 23.98 23.5 22.97 15 138 (2207) 17.67 17.6 Measurement of Pore Size Distribution Total 17.57 porosity is a parameter that does not adequately describe the 16 124 (1988) 25.84 25.5 pore characteristics of a concrete. A fuller specification 25.06 17 114 (1818) 25.57 25.5 should incl ude pore size, shape, and structure. So that the 25.46 pore size distribution of the various concretes could be 18 97 (1551) 39.04 39.0 investigated thoroughly, it was necessary to prepare 2-in. 39.05 (50-mm) mortar cubes using the fine aggregate materials. 19 140 (2238) 13.09 12.9 The pore size of the voids in the mortar samples was 12.73 20 140 (2243) 12.49 13.1 determined by mercury intrusion porosimetry (Rootare 13.72 1970; Bakel et al. 1981). This method involves evacuating 21 109 (1745) 25.86 26.1 the gas from the pores and then forcing mercury into a 26.24 sample by gradually increasing the pressure on the mercury. Both the volume of mercury intruded and the pressure to achieve the intrusion were measured. From these data and dried at 221°F (l05°C) for 24 hours to achieve constant the knowledge of the wetting angle and surface tension of weight. Coarse aggregates of various sizes representing the mercury (130° and 2.8 X 1O-4Ib/in. [48.4 dynes/em]), each aggregate type were used to determine the pore size respectively, the total porosity of the specimen may be distribution of the aggregate phase in different concrete determined. The cumulative pore volume against pore mixes. The mercury intrusion porosimetry results for both diameter or a pore size distribution curve may then be the mortars and coarse aggregates are given in Table 5. plotted. The instrument used in this investigation was a From the data obtained from the mortar samples and mercury intrusion porosimeter capable of exerting 60,000 coarse aggregates separately, it was possible to calculate the Ib/in.2 (414 MPa) pressure. Samples 1 in. (25 mm) in median pore diameter of the concretes. These are also diameter were cored from the mortar cubes using a dia­ shown in Table 5. A summary of the results is given in mond saw. The cored specimens were then cut to 3/8-in. to Table 6, which also includes the measured values of 1/2-in.- (10- to 12-mm)-thick samples for testing. Before thermal conductivity corrected to 3 % moisture content by determining their pore size distribution, the samples were volume.

93 TABLES TABLE 6 Mercury Intrusion Porosimetry Results Thermal Conductivity, Dry Density, Porosity, for Mortars and Coarse Aggregates and Median Pore Diameter for the Concretes Studied

% Mi~ Measured Mean Dry M'~ Median MuoJeleu Difference Mortar Relative Total Calculated No A@3%mc DenSi¥ Total Pore Values of from Code Density Intrusion Concrete Ihlft. Porosity Diameter A at ]'il, Measured 3 (Micron) Values Name (glee) Volume Median Pore ~/~~iImK (kglm ) (%) me b Vol

(cc/g) Diameter I 2.76 143 (Z2S2) 12.9 0.33 2.48 -10.1 (2091) 14.0 0.48 1.94 -10.6 (~m) 2 2.17 131 3 1.75 141 (2248) 15.0 G.20 1.76 0.0 4 1.50 127 (2033) 21.0 0.27 1.32 -12.0 NO.53 2.4552 0.0807 0.0936 5 0.91 116 (1858) 21.0 O.:U 1.04 +14.2 6 0.65 97 (1557) 38.5 0,82 0.73 +12.3 AE/O.43 2.4468 0.1079 0.3759 7 2.56 141 (2250) 13.9 0.35 2.34 -8.2 NO.6 2.4473 0.0843 0.1367 8 1.73 141 (2261) 14.1 0.23 1.95 +12.7 , 0.65 98 (1570) 23.4 0.91 0.57 -12.3 AE/O.5 2.4218 0.1219 0.2581 10 2.86 146 (2341) 10.2 0.39 2.99 +4.5 PNO.9 2.4698 0.1189 0.2790 II 1.89 143 (2291) 12.1 0.23 2.15 +13.7 PAE/O.8 2.4038 0.1307 0.2823 12 0.69 103 (1647) 25.1 0.97 0.61 . [1.5 13 2.61 140 (2236) 15.1 0.50 2.47 -5.4 NO.8 2.4698 0.0892 0.1348 14 1.99 126 (2020) 23.5 0.82 1.72 -13.6 PNO.8 2.4018 0.0827 0.1367 15 L81 138 (2207) 17.6 0.28 1.78 -1.6 16 1.44 124 (1988) 25.5 1.11 [.47 ... 2.0 BIO.8 2.4272 0.1006 0.1649 17 0.82 (1818) 25.5 0.87 0.93 +13.4 PB/O.8 2.4512 0.0789 0.1175 18 0.63 "'97 (1551) 39.0 1.32 0.57 -9.5 B/O.56 2.4580 0.0820 0.0917 19 2.70 140 (2238) 12.9 0.45 2.53 -6.3 20 1.96 140 (2243) 13.1 0.29 2.19 +11.7 BIO.66 2.4408 0.0868 0.1693 21 0.85 109 (1745) 26.4 0.80 0.96 +12.9 PB/1.05 2.4005 0.1093 0.3394 C/O.9 2.4881 0.1088 0.3233 CE/O.8 2.4240 0.l290 1.0057 CIO.95 2.4601 0.1213 0.3752 linking only one concrete property to thermal conductivity CE/O.9 2.5068 0.1418 1.8138 are adequate for a narrow range of concretes made with the PCl1.29 2.5168 0.0995 0.2917 same aggregate and cement phases. To develop a model that PCE/1.29 2.4552 0.1208 1.2760 predicts thermal conductivity to an acceptable accuracy over C/O.8 2.4.588 0.0944 0.3003 a wide range of concretes requires that it be based on the PC/O.S 2.4552 0.0807 0.0938 parameters that influence the overall heat transfer process. Consequently, the interrelated data reported in this paper Notes: were used to develop a model that included the pore P mortar sieved from Pellite characteristics and the density, which, in this case, is a A,B Cement-sand ratio of 1:2.33 numerical characterization of the solid phases. C Cement-sand ratio of 1:3.73 The data were processed using a statistical multivariate E Air-entrained mix analytical technique that resulted in the development of a polynomial expression of the following form:

ANALYSIS OF RESULTS AND 1. 77 X 1O-6p2 -1139.67(MPD/P)2_6.08 X 1O-3p DEVELOPMENT OF A NEW MODEL -54.21(MPD/P)+6.50 X 1O-2p(MPD/P) +5.47; r 0.962, level of significance 99.999%. It has been found that a satisfactory way to quantify the pore size distribution of a concrete is to characterize the where porous phase by total porosity and a calculated median pore A thermal conductivity, W/m'K; diameter. The specification of the solid phase only included p dry density, kg/m3; the ratios of aggregates to cement, since their intluence was P total porosity, %; reflected directly in changes of porosity and median pore MPD median pore diameter, ftm. diameter. Increasing the percentage of quartz or limestone reduced the total porosity of the concrete because both The values of thermal conductivity predicted by the aggregates had less pore volume than the hydrated cement above equation are given in Table 6 together with the and sand. Increasing the volume of pellite increased the percentage difference from their respective measured value_ total porosity and the median pore diameter. The effect of It can be seen that the maximum difference is equal to air entrainment was to increase the total porosity and the A±14%. median pore diameter. The changes in density are the result Further analysis revealed that the variables that can be of the combined effects of composition and pore volume. used to predict the thennal conductivity of a concrete with The mechanisms by which heat is transferred through a 97% confidence limit are density, porosity, and median a concrete are an interrelated function of the properties of pore diameter. Density is the most important variable in the both the solid phase and the porosity. Empirical relations relationship and porosity the next.

94 The above model contains a term involving median unguarded hot-plate method. UDC 536.212'(X114. pore diameter, which is a difficult parameter to obtain London: British Standards Institution. without sophisticated equipment and trained personnel and BSI. 1986. British Standard 882, Specification for aggre­ is the least significant in the relationship. For practical gates from natural sources for concrete. London: purposes, a less accurate model relating thermal conduc­ British Standards Institution. tivity solely to density and total porosity is being developed. Campbell-Allen, D., and C.P. Thome. 1963. The thermal conductivity of concrete. Magazine of Concrete Re­ CONCLUSIONS search 15(43): 39-48. Eucken, A.A. 1932. Forch Gebiete Ingenieurw. B3, VDI­ 1. Porosity and median pore diameter are parameters that Forschingsheft 353, 11(6): 940. satisfactorily represent changes in the composition and Ganjian, E. 1990. The relationship between porosity and mineralogy of concrete and have been shown to be thermal conductivity of concrete. Unpublished Ph.D. important variables that can be used to predict thermal thesis, Department of Civil Engineering, University of conductivity. Leeds, UK. 2. The highly significant statistical model relating thermal Maxwell, J.C. 1892. A treatise on electricity and mag­ conductivity to the characteristics of concrete as netism, 3d ed., vol. I, 440. Oxford: Clarendon Press. represented by density, porosity, and median pore Pratt, A.W. 1962. Heat transfer in porous material. Re­ diameter presented in this paper predicts values of A search 15(5): 214-244. within ± 14% of the value measured to BS 874 using Reynolds, J.A., and J.M. Hough. 1957. Proc. Soc. B70: a plain hot-plate apparatus. 769. 3. Using the model developed, a concrete sample could be Rootare, H.M. 1970. Review of mercury porosimetry. taken from an existing structure, its dry density and Advanced Experimental Technique in Powder Metallur­ porosity parameters determined, and then its thermal gy 5: 225-252. conductivity predicted. Simpson, A., and A.D. Stuckes. 1986. Thermal conduc­ tivity of porous materials; 1, theoretical treatment of REFERENCES conduction processes. BSER&T 7(2): 78-86. Tinker, J.A. 1987. Modelling the thermal conductivity of Bakel, J.V., S. Modry, and M. Savata. 1981. Mercury mUlti-phase materials containing moisture. Numerical porosimetry: State of the art. Powder Technology 29: Methods in Thermal Problems 5(1): 669-680. 1-12. Tinker, J.A., J.G. Cabrera, and E. Ganjian. 1989. Thermal Brailsford, A.D., and K.G. Major. 1964. The thermal transfer in masonry materials-Moisture correction conductivity of aggregates of several phases, including factors. Proc. XI Int. Congress CIB 89, theme I, vol. porous material. British Journal ofApplied Physics 15: III, 427-435. 313. Valore, R.C. 1980. Calculation of U-values of hollow BSI. 1988. British Standard 874, Part 2, Section 2.2, concrete masonry. American Concrete Inst. 2(2): 40- Methods for determining thermal insulating properties; 63.